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Math Help - Linear Algebra true or fasle statements

  1. #1
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    Linear Algebra true or fasle statements

    are these statements true or false? give a reason or a counterexample to justify your answers.
    (a) A square invertible matrix A with complex number entries is diagonalizable just when inverse of A is diagonalizable.
    (b)Every diagonalizable matrix whose entries are complex numbers is normal.
    (c)Every square matrix has a kernel dimension 0.
    (d)There is a 3x3 matrix A with real number entries so that the kernel of A is 2-dimensional and the kernel of A-I is also 2-dimensional.
    (e)A square invertible matrix A with real number entries is diagonalizable just when inverse of A is diagonalizable.
    (f)If A is a normal matrix with complex number entries and A^2=0 then A=0.
    (g)Any square matrix with complex number entries is diagonalizable.
    (h)Suppose that A is an nxn matrix with real number entries. Suppose that Ax has the same ength as x, for every vector x in R^n. Then A is orthogonal.
    (i)Suppose that A is symmetric matrix. The eigenvalues of A^2 are the squares of the eigenvalues of A.
    (j)Every diagonalizable matrix whose entries are real numbers is symmetric.
    (k)If a square matrix A id diagonalizable, then A^2 is diagonalizable.
    (l)Every invertible matrix has kernel of dimension 0.
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  2. #2
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    Re: Linear Algebra true or fasle statements

    Think of some examples satisfying the premises of these propositions. Do all of them satisfy the conclusion? If not, then you have a counterexample. If so, do you know why?
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